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Unformatted text preview: 6.042/18.062J Mathematics for Computer Science May 10, 2005 Srini Devadas and Eric Lehman Lecture Notes Random Walks 1 Random Walks A drunkard stumbles out of a bar. Each second, he either staggers one step to the left or staggers one step to the right, with equal probability. His home lies x steps to his left, and a canal lies y steps to his right. This are several natural questions, including: 1. What is the probability that the drunkard arrives safely at home instead of falling into the canal? 2. What is the expected duration of his journey, however it ends? The drunkards meandering path is called a random walk . Random walks are an im- portant subject, because they can model such a wide array of phenomena. For example, in physics, random walks in three-dimensional space are used to model gas diffusion. In computer science, the Google search engine uses random walks through the graph of web links to determine the relative importance of websites. In finance theory, random walks can serve as a model for the uctuation of market prices. And, in this lecture, well explore some more palatable applications. 2 Pass the Candy Pass the Candy is game involving n students, one professor, and a bowl of candy. The students are numbered 1 to n , and the professor is numbered 0. Everyone stands in a circle as shown below. 1 2 3 n/2 n-2 n-1 n (professor) 2 Random Walks Initially, the professor has the candy bowl. He withdraws a piece of candy and then passes the bowl either left or right, with equal probability. Each person who receives the bowl thereafter does the same thing: he or she takes a piece of candy from the bowl and then passes it either left or right, with equal probability. In effect, the bowl goes for a random walk around the circle of players. The last person to receive a piece of candy is declared the winner and gets to keep the entire bowl. Which player is most likely to win? A natural guess is player n/ 2 . She seems most likely to receive the bowl last and thus to win the game, because she is farthest from the professor. On the other hand, players 1 and n seem almost certain to receive the bowl far too early in the process to win the game. Lets see if this intuition is right or wrong! 2.1 A Simpler Problem Lets begin by looking at a simpler problem. Suppose that the players are arranged in a line, rather than a circle: A S 1 S 2 . . . S k B candy The players are named A , S 1 , S 2 , . . . , S k , and B , as shown. Initially, player S 1 has the candy bowl. As before, whenever a player gets the bowl, he or she takes a piece of candy and then passes the bowl either left or right, with...
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